Modelleerimine ja Juhtimine Tehisnärvivõrgudega

1. Modelleerimine ja Juhtimine Tehisnärvivõrgudega Identification and Control with artificial neural networks (2nd part) Eduard Petlenkov, Automaatikainstituut Eduard Petlenkov, Automaatikainstituut, 2015

2. Supervised learning Supervised learning is a training algorithm based on known input-output data. X is an input vector; Ypis a vector of reference outputs corresponding to input vector X Y is a vector of NN’s outputs corresponding to input vector X NNis a mathematica function of the network (Y=NN(X)) Eduard Petlenkov, Automaatikainstituut, 2015

3. Õpetamine (2) • Computation of NN’s outputs corresponding to the current NN’s paraneters; • Computation of NN’s error; • Updating NN’s parameters by using selected training algorithm in order to minimize the error. Network training procedure consists of the following 3 steps: Eduard Petlenkov, Automaatikainstituut, 2015

4. Gradient descent error backpropagation (1) Error function: - NN’s output -inputs used for training; - NN’s function; - etalons (references) for outputs=desired outputs. function to be minimized: Eduard Petlenkov, Automaatikainstituut, 2015

5. Gradient descent error backpropagation (2) Eduard Petlenkov, Automaatikainstituut, 2015

6. Global minimum problem Eduard Petlenkov, Automaatikainstituut, 2015

7. Training of a double-layer perceptron using Gradient descent error backpropagation (1) i - number of input; j – number of neuron in the hidden layer; k – number of output neuron; NN’s input at time instance t; ; - Values oh the hidden layer weighting coefficients at time instance t; - Values oh the hidden layer biases at time instance t; - Activation function of the hidden layer neurons - Outputs of the output layer neurons at time instancet; - Values oh the output layer weighting coefficients at time instance t; - Values oh the output layer biases at time instance t; - Activation function of the output layer neurons; - Outputs of the network at time instance t; Eduard Petlenkov, Automaatikainstituut, 2015

8. Training of a double-layer perceptron using Gradient descent error backpropagation (2) error: gradients: Signals distributing information about error from output layer to previous layers (that is why the method is called error backpropagation): Are derivatives of the corresponding layers activation functions. Eduard Petlenkov, Automaatikainstituut, 2015

9. Training of a double-layer perceptron using Gradient descent error backpropagation (3) Updated weighting coefficients: learning rate Eduard Petlenkov, Automaatikainstituut, 2015

10. Gradient Descent (GD) method vs Levenberg-Marquardt (LM) method - Gradient - Hessian GD: LM: Eduard Petlenkov, Automaatikainstituut, 2015

11. Näide MATLABis (1) P1=(rand(1,100)-0.5)*20% Juhuslikud sisendid. 100 väärtust vahemikus [-10 +10] P2=(rand(1,100)-0.5)*20 P=[P1;P2]% Närvivõrgu sisendite moodustamine T=0.3*P1+0.9*P2 % Vastavate väljundite arvutamine net=newff([-10 10;-10 10],[5 1],{'purelin' 'purelin'},'traingd') % Närvivõrgu genereerimine teinesisend [min max] neuronite arvud igas kihis Viimase kihi neuronite arv = väljundite arv esimene sisend [min max] Teise kihi neuronite aktiveerimisfunktsioon Esimese kihi neuronite aktiveerimisfunktsioon õppimisalgoritm net.trainParam.show=1; Treenimise parameetrid net.trainParam.epochs=100;

12. Näide MATLABis (2) net=train(net,P,T) % Närvivõrgu treenimine out=sim(net,[3 -2]) % Närvivõrgu simuleerimine Teise sisendi väärtus Esimese sisendi väärtus W1=net.IW{1,1} % esimese kihi kaalukoefitsientide maatriks W2=net.LW{2,1} % teise kihi kaalukoefitsientide maatriks B1=net.b{1} % esimese kihi neuronite nihete vektor B2=net.b{2} % teise kihi neuronite nihete vektor Eduard Petlenkov, Automaatikainstituut, 2015